vol. 174, supplement
the american naturalist
july 2009
On the Causes of Selection for Recombination
Underlying the Red Queen Hypothesis
Marcel Salathé,1,*,† Roger D. Kouyos,2,† and Sebastian Bonhoeffer2
1. Department of Biology, Stanford University, Stanford, California 94305; 2. Institute of Integrative Biology, ETH-Zürich (Swiss
Federal Institute of Technology), CH-8092 Zürich, Switzerland
abstract: The vast majority of plant and animal species reproduce
sexually despite the costs associated with sexual reproduction. Genetic
recombination might outweigh these costs if it helps the species
escape parasite pressure by creating rare or novel genotypes, an idea
known as the Red Queen hypothesis. Selection for recombination
can be driven by short- and long-term effects, but the relative importance of these effects and their dependency on the parameters of
an antagonistic species interaction remain unclear. We use computer
simulations of a mathematical model of host-parasite coevolution to
measure those effects under a wide range of parameters. We find
that the real driving force underlying the Red Queen hypothesis is
neither the immediate, next-generation, short-term effect nor the
long-term effect but in fact a delayed short-term effect. Our results
highlight the importance of differentiating clearly between immediate
and delayed short-term effects when attempting to elucidate the
mechanism underlying selection for recombination in the Red Queen
hypothesis.
Keywords: recombination, evolution of sex, Red Queen hypothesis,
fluctuating epistasis, short-term effect, long-term effect.
Introduction
Why is it that, in almost all animal and plant species,
offspring individuals have different combinations of genes
from their parents? While the proximate cause of this phenomenon, genetic recombination, has been worked out in
great detail, its ultimate cause continues to remain enigmatic (see Otto 2009). Numerous hypotheses have been
proposed to explain the evolutionary maintenance of recombination. Because the only direct effect of genetic recombination is the reduction of statistical associations
among genes (i.e., linkage disequilibria, [LD]), any hypothesis that aims to explain the selective benefit of recombination must explain (i) why breaking up genetic
associations is beneficial and (ii) by which process such
genetic associations are continually recreated.
* Corresponding author; e-mail: [email protected].
†
These authors contributed equally to this article.
Am. Nat. 2009. Vol. 174, pp. S31–S42. 䉷 2009 by The University of Chicago.
0003-0147/2009/1740S1-50689$15.00. All rights reserved.
DOI: 10.1086/599085
One possible benefit of recombination is that the breaking up of genetic associations may result in an increase in
genetic variance, which may in turn increase the response
to selection. The problem with this argument is that recombination increases genetic variance only when genotypes with intermediate fitness are overrepresented in a
population (or, in mathematical terms, when LD ! 0). If
the genotypes with high or low fitness are overrepresented
(i.e., LD 1 0), recombination will result in a decrease of
genetic variance, which in turn hampers the response to
selection.
Two processes have been identified as potential causes
of negative LD: negative epistasis among deleterious alleles
and drift in conjunction with selection. Negative epistasis
among deleterious alleles implies that alleles have a stronger fitness-reducing effect in combination than is expected
from their effect in isolation. Whenever combinations of
deleterious alleles are more unfit than expected (negative
epistasis), it is intuitive that these combinations become
less frequent after selection than is expected on the basis
of the frequency of the individual alleles. Hence, negative
epistasis acts as a force that generates negative LD. Drift
refers to the change of allele frequencies in finite populations due to chance events, and while drift can cause
both negative and positive LD, positive LD disappear faster
by selection than negative LD. The net result of this process
is a prevalence of negative LD, an effect commonly known
as the Hill-Robertson effect (Hill and Robertson 1966).
Speeding up selection by increasing variance is a process
that takes some time: first, the associations must be broken
down, releasing beneficial alleles from their deleterious
backgrounds, and then those alleles must spread in the
population, dragging alleles that increase rates of recombination with them. This effect, known as the long-term
effect, is not, however, the only effect of recombination
(Barton 1995). By breaking down genetic associations, recombination also has an effect that is immediate: it creates
new combinations of alleles that might be more or less fit
than the combinations of alleles in the previous generation.
This effect is known as the short-term effect, because it
becomes manifest immediately in the next generation. The
S32 The American Naturalist
effect is generally thought to be detrimental in systems at
selection-mutation(-drift) balance. The reason is that the
combinations of alleles that are overrepresented in a population have been favored by selection in the past, and
thus breaking them down typically results in a population
whose average fitness is lower.
The net selective effect on any gene that influences the
rate of recombination is thus a result of two effects, the
short-term and the long-term effect. In models that are
based on simple fitness landscapes that are constant over
time, it is well understood under which circumstances the
two effects lead to a net benefit for recombination (for a
review, see Otto and Lenormand 2002). In such models,
the short-term effect generally tends to be negative, because recombination breaks down the overly fit and thus
overly frequent allele combinations. Provided that the population is sufficiently large that epistasis is the dominating
force generating LD, the long-term effect is beneficial only
when epistasis (and thus LD) is negative. The long-term
effect can outweigh the short-term effect if epistasis is
negative but sufficiently weak.
The Red Queen hypothesis. The assumption that fitness
landscapes are constant over time is overly simplistic for
many biological scenarios. Thus, an alternative hypothesis
to explain the ubiquity of genetic recombination is that it
may continually create novel genotypes that are at a selective advantage in an ever-changing environment. The
idea that recombination might be beneficial in changing
environments can already be found in the writings of
Wright, Muller, and others (Muller 1932; Wright 1932;
Sturtevant and Mather 1938; Haldane 1949). Initially, there
was some doubt as to whether environments change fast
enough to cause selection for recombination (Charlesworth 1976; Bell 1982), but many theoretical studies have
shown in the meanwhile that host-parasite coevolution, a
process that likely affects almost all species, provides the
necessary conditions for such a mechanism to work (for
a review, see Salathé et al. 2008b). Because parasites adapt
quickly to become better at infecting the most frequent
host genotype, recombination in the host might be beneficial because it creates rare or novel genotypes that are
at a selective advantage. This idea is nowadays known as
the Red Queen hypothesis (RQH; Bell 1982).
It is worth revisiting the conditions under which the
RQH works. First, the outcome of an interaction between
a host and a parasite must be at least in part genetically
determined, an assumption that is likely to be met frequently. Second, selection must be strong on at least one
of the coevolving species (Salathé et al. 2008a). There has
been considerable debate about whether parasites are generally virulent enough to cause enough selection on hosts
for recombination to be beneficial (Peters and Lively 1999;
Schmid-Hempel and Jokela 2002; Otto and Nuismer
2004). However, it is plausible to assume that parasites
often do experience substantial selection from the host,
because many parasites cannot reproduce at all if they fail
to infect a host (or reproduce at much slower rates outside
of a host than within a host). Moreover, even if selection
on the parasite is relatively weak, the fact that most parasites have much shorter generation times than their hosts
implies that they can adapt much faster than their hosts,
effectively increasing the selective pressure on hosts (Salathé et al. 2008a). Third, the interactions between host
loci mediating parasite resistance must be epistatic (Kouyos et al. 2007) or display dominance (Agrawal and Otto
2006). In the absence of epistatic interactions among loci,
strong LD of constant sign build up quickly and cause
strong selection against recombination (Kouyos et al.
2009). Data from plant and animal studies suggest that
epistatic interactions among disease-resistance loci are indeed prevalent (Kover and Caicedo 2001).
Thus, while theoretical studies have clarified the role of
both the strength of selection and the genetic interaction
system necessary for the RQH to work, the exact causes
of selection for recombination remain to be fully investigated. Three recent studies have shed some light on this
issue. Peters and Lively (2007) investigated the spread of
a gene for recombination in an initially nonrecombining
population and found that recombination provides an
“immediate (next-generation) fitness benefit and a delayed
(two or more generations) increase in the rate of response
to directional selection” (p. 1206). Gandon and Otto
(2007) analyzed a model where allele frequencies remain
fixed over time (the so-called Nee model [Nee 1989]),
which has the benefit that directional selection is absent
and a long-term effect based on increased genetic variance
can thus be excluded. Their analysis focused on the time
lag between epistasis and LD fluctuations, arguing that
this lag yields a short-term benefit for recombination in
generations where LD and epistasis have opposite signs.
Finally, we argued (Salathé et al. 2008b) that the shortterm effect should be subdivided into an immediate effect
and a delayed effect. Using the Nee model, we demonstrated that the immediate short-term effect is often detrimental while the delayed short-term effect is generally
beneficial for recombination. In particular, we argued that
the immediate short-term effect, approximated by the
product of epistasis and LD, has almost no predictive
power with respect to the evolution of recombination.
Disentangling the various effects affecting the selection
pressure on a recombination modifier is important both
theoretically and empirically. For example, if the immediate short-term effect in the next generation does not
have any power to predict the direction of selection for
recombination, then measuring this effect will not be useful for demonstrating the role of the RQH in natural pop-
Causes of Selection for Recombination S33
ulations. Our previous analysis (Salathé et al. 2008b) offered only limited insight into the mechanism underlying
the selection for recombination in the RQH, because it
was based on the somewhat unrealistic Nee model and
was restricted to a small parameter space. In this study,
we extend our previous work by expanding the parameter
space and by presenting an approach that allows the disentanglement of the short-term effect (both immediate
and delayed) from the long-term effect in a model where
both the short-term and long-term effects are present. We
find that neither the long-term effect nor the immediate
short-term effect have much power to predict whether
recombination is favored or disfavored in Red Queen
models. In fact, for most of the parameter space investigated, the two effects are either irrelevant (i.e., of negligible
magnitude) or quite often misleading (i.e., pointing in
opposite directions) if looked at in isolation. We identify
the delayed short-term effect as the best predictor for the
evolution of recombination in the RQH.
Material and Methods
We use a frequency-based deterministic model that has
been described elsewhere (Kouyos et al. 2007) to simulate
host-parasite coevolution. Briefly, hosts and parasites are
haploid and have two loci with two possible alleles (0 and
1) that determine the outcome of an interaction between
host and parasite, depending on the interaction model
used (see below). Hosts possess an additional modifier
locus with two possible alleles, m (wild type) and M (mutant), that define the rate at which hosts recombine. The
modifier locus is positioned at one end of the genome.
Thus, there are eight possible host haplotypes (00m, 00M,
01m, 01M, 10m, 10M, 11m, and 11M) and four possible
parasite haplotypes (00, 01, 10, and 11).
A simulation runs for t time steps, which corresponds
to t host or parasite generations. Hosts and parasites are
assumed to have the same generation time. All results
presented below are from simulations that ran for 2,000
host generations. At each time step, the following processes
occur: host reproduction, host selection, host mutation,
parasite reproduction, parasite selection, and parasite mutation. The recursion equations for host and parasite genotype frequencies are as follows:
P
t⫹1
f
P
H
P
p RSM (ft , ft ),
(1)
H
ft⫹1
p RSMH(ft H, ft P ),
(2)
where the superscripts H and P denote the host and parasite populations, respectively, and the function RSM denotes the successive actions of reproduction, selection, and
mutation. We now describe these three processes in detail.
We assume that parasites reproduce clonally while hosts
reproduce by recombining their genotypes. Recombination occurs at a rate that depends on the alleles present
at the modifier locus of two recombining genotypes. The
recombination rate between the interaction loci is rbaseline
if both recombining genomes carry allele m. It is increased
by a small amount, Dr or 2Dr, if one or both genomes,
respectively, carry allele M. The recombination rate between the modifier and the interaction loci is rmodifier (only
one recombination rate has to be defined in this case,
because a recombination event between the modifier locus
and the interaction loci has an effect only if the modifier
alleles of the two recombining genomes are different). Because we are interested in the fate of the allele M increasing
the recombination rate, Dr 1 0 in all simulations.
Selection is determined by a fitness matrix, as defined
by the interaction model chosen (see below and Salathé
et al. 2008b, box 1), and by the genotype frequencies of
the coevolving species. More explicitly, the fitness matrices
W H p (wijH)4#4 and W P p (wijP )4#4 define the fitness of
each possible interaction of host genotype i and parasite
genotype j in a two-locus/two-allele system. Assuming that
the probability of interaction between host genotype i and
parasite genotype j is the product of their respective frequencies, fi H and fj P, the frequency of host genotype i after
selection is
fi H p fi H
wiH
,
冘k wkH fkH
(3)
where wiH is the fitness of host genotype i, given by
wiH p
冘
wijH fj P.
(4)
j
The parasite frequencies after selection, f P, are calculated
analogously.
Finally, mutations at the interaction loci occur at a rate
of 10⫺5 per locus per generation in both hosts and parasites. If a mutation occurs, allele 0 becomes allele 1, and
vice versa. There is no mutation at the modifier locus.
Interaction Model
We use two different interaction models: the Nee model
and the matching-allele model (MA model). The MA
model is widely used in the context of the RQH, and in
the two-locus/two-allele setup used here, the MA model
assumes that a host genotype is susceptible to a given
parasite genotype if and only if the alleles at both interaction loci match. In that case, the fitness of the host
genotype is 1 ⫺ s H and the fitness of the parasite genotype
is 1. In all other cases (i.e., where the host and parasite
S34 The American Naturalist
Figure 1: Disentangling the effects of a recombination modifier. A recombination event is indicated by the thick dotted line. In the immediately
following generation, g0, only the immediate short-term effect (A) is causing selection for or against recombination. After that (gn 1 0), two types of
effects can be responsible for selection: the delayed short-term effect (B), which is a continuation of the short-term effect, and the long-term effect
(C) based on directional selection. Because these two effects occur at the same time, it is important to distinguish between them clearly. The total
short-term effect is given by lines A and B.
genotype do not fully match), the host is assumed to be
fully resistant, and thus the fitness of the host genotype is
1 and the fitness of the parasite genotype is 1 ⫺ s P.
The Nee model assumes that the host genotype is resistant to a given parasite genotype if and only if the alleles
at exactly one interaction locus match; for example, the
host genotype 01 is resistant to parasite genotypes 11 and
00 but not to genotypes 01 and 10. In the case of resistance,
the fitness of the host genotype is 1 and the fitness of the
parasite genotype is 1 ⫺ s P. In all other cases, the fitness
of the host genotype is 1 ⫺ s H and the fitness of the parasite
genotype is 1. The Nee model lacks biological realism but
has the useful property that the allele frequencies converge
to 0.5. Constant allele frequencies imply that directional
selection is absent; therefore, after a transient burn-in
phase, directional selection can be neglected, and thus
there is no long-term effect selecting for recombination
in this model.
Simulations
Both host and parasite populations are assumed to be
infinite in size, and they are initiated with random genotype frequencies, except at the modifier locus, where
only allele m is initially present (i.e., recombination occurs
initially at a rate rbaseline). In order to investigate the fate
of the modifier allele (M) that increases recombination,
we introduce the M allele in 0.01% of the population, after
a burn-in phase of 1,000 host generations. We then let the
simulation run for another 1,000 host generations, during
which we measure a variety of quantities, such as epistasis,
linkage disequilibrium, and the frequency of the modifier
allele M.
The effect of altering linkage disequilibrium on the
mean fitness of offspring depends on epistasis measured
on an additive scale. Epistasis in the host population is
thus measured as
␧H p
w00H ⫹ w11H ⫺ w01H ⫺ w10H
,
w̄ H
(5)
where wiH is the fitness of host genotype i, as defined in
equation (4). Linkage disequilibrium in the host is measured as
LD H p f00H f11H ⫺ f01H f10H.
(6)
Measuring Short- and Long-Term Effects
In order to disentangle the various effects of selection acting on a recombination modifier, we aim to quantify the
short- and long-term effects on the recombination modifier allele M. The short-term effect acts in the generation
immediately after a given recombination event (fig. 1, line
A), but it is not limited only to that generation (fig. 1,
line B). The long-term effect of a given recombination
event begins to act one generation after the event. In the
generation immediately after the recombination event, genetic associations are broken down, but only in the subsequent generations does the increased (or decreased) variance have a selective effect on the modifier (fig. 1, line
C).
Causes of Selection for Recombination S35
In the Nee model, where directional selection is absent,
we can directly measure the short-term effect acting immediately in the next generation (the so-called immediate
short-term effect) and the short-term effect acting after
one generation (the so-called delayed short-term effect).
In essence, we measure the short-term effect by comparing
populations that differ in their recombination rates only
in a single generation. More precisely, we implement the
following procedure. After each generation of a simulation,
we pause the simulation and create four test populations
(two host populations H1 and H2, and two parasite populations P1 and P2), which we initiate with the current
host and parasite frequencies of the main populations. We
then let the test populations coevolve for 50 host generations (H1 with P1 and H2 with P2), with the only difference being that for the first generation in population
H2, Dr is set to 0 (i.e., the modifier has no effect on the
recombination rate). After this first generation, Dr is set
back to its original value. We then measure selection acting
on the modifier n generations after the first generation
(sn) as the difference between the mean fitness of the genotypes carrying allele M in populations H1 and H2 divided by the total mean fitness in the (n ⫹ 1)th generation
after the initiation of the test populations:
sn p
¯ (n ⫹ 1) ⫺ w
¯ (n ⫹ 1)
w
.
w̄MH1(n ⫹ 1)
H1
M
H2
M
(7)
The value of s0 corresponds to the immediate short-term
effect, while the sum of all sn values, where n 1 0, corresponds to the delayed short-term effect.
In the MA model, we use the same procedure to measure
the immediate short-term effect, but because the delayed
short-term effect and the long-term effect act at the same
time, it is not possible to differentiate between these two
effects with the method described above. Instead, we use
a mathematical approximation to calculate the strength of
selection acting on the modifier due to short- and longterm effects. The short-term effect is due to differences in
LD of the M-linked and m-linked genotypes, while the
long-term effect is due to differences in allele frequencies
of the M-linked and m-linked genotypes. Mathematically,
the strength of selection on the modifier can be expressed
as
sM p
¯M ⫺ w
¯m
w
,
w̄
(8)
¯ M and w
¯ m denote the mean fitness of genotypes
where w
with modifier alleles M and m, respectively. The mean
fitness is a function of the allele frequencies and LD, that
¯ p w(p
¯ 1, p2 , LD), where pi denotes the allele frequency
is, w
at interaction locus i. For a weak modifier, sM can be approximated as
sM p
p
¯M ⫺ w
¯ m)
(w
w̄
¯ 1M, p2M, LD M) ⫺ w(p
¯ 1m, p2m, LD m)
w(p
w̄(p1, p2 , LD)
¯ 1m, p2m, LD m)
p [(p1M ⫺ p1m)⭸p 1w(p
¯ 1m, p2m, LD m)
⫹ (p2M ⫺ p2m)⭸p 2w(p
(9)
¯ 1m, p2m, LD m)]
⫹ (LD M ⫺ LD m)⭸LDw(p
¯ 1, p2 , LD),
/w(p
where piM and LDM denote allele frequencies and LD, respectively, of the subpopulation with allele M and pim and
LDm denote allele frequencies and LD, respectively, of the
subpopulation with allele m. The first two terms correspond to the allele-frequency-dependent component of sM,
whereas the third term corresponds to the LD-dependent
component of sM. Hence,
s long p
sshort p
¯ ⫹ (p2M ⫺ p2m)⭸p 2w
¯
(p1M ⫺ p1m)⭸p 1w
w̄(p1, p2 , LD)
¯
(LD M ⫺ LD m)⭸LDw
.
w̄(p1, p2 , LD)
,
(10)
(See the appendix for explicit expressions of the shortand long-term effects.)
Results
Our first goal was to extend our previous analysis of the
short-term effect in the Nee model, where, because of the
absence of directional selection, there is no long-term effect. We previously argued that although the short-term
effect is manifest immediately in the next generation, it is
also manifest in subsequent generations (i.e., with a delay).
Moreover, we showed that the immediate short-term effect
can select against recombination while the delayed shortterm effect selects for recombination (Salathé et al. 2008b).
In particular, the immediate short-term effect is usually
much weaker than the delayed short-term effect and thus
has almost no predictive power with respect to the evolution of recombination. Figure 2 extends those results
into the entire parameter space of possible selection
strengths acting on both host (sH, i.e., virulence) and parasite (sP).
Figure 2A shows selection on a modifier causing in-
S36 The American Naturalist
Figure 2: Results from the Nee model. Red to yellow: selection for recombination; blue to black: selection against recombination. A, Change of
modifier frequency; B, linkage disequilibrium # epistasis; C, immediate short-term effect; D, delayed short-term effect. The short-term effects are
measured as described in “Material and Methods.” Parameters: rbaseline p 0.0025; Dr p 0.0005; rmodifier p 0.5.
creased recombination. Such a modifier is selected when
at least one of the coevolving species experiences sufficient
selection, confirming similar findings in the matchingallele model (Salathé et al. 2008a). As expected, the product
of LD and epistasis (fig. 2B) and the immediate shortterm effect (fig. 2C) are qualitatively equivalent but opposite in sign: a negative product corresponds to a positive
immediate short-term effect, and vice versa. However, the
immediate short-term effect (fig. 2C) is weak compared
to the effective selection on the modifier (fig. 2A), in stark
contrast to the delayed short-term effect, which is much
stronger (fig. 2D). It is the sum of the immediate and
delayed short-term effects that is acting on the modifier,
but as can be seen from figure 2C and 2D, the immediate
effect is largely irrelevant quantitatively and can be misleading qualitatively.
Epistasis and LD for a certain time span of a typical
simulation run are shown in figure 3A. Figure 3B shows
the short-term effect measured with a delay of 0 (solid
line), 1 (dashed line), and 2 (dotted line) generations; thus,
the solid line shows the immediate short-term effect
(n p 0). Delayed short-term effects (n 1 0) show positive
selection more frequently than the immediate effect. The
reason for this is that the immediate and the delayed shortterm effects become negative at the same time but the
delayed short-term effect becomes positive earlier than the
immediate short-term effect (around time steps 8 and 24
in fig. 3B). Why is this so? The delayed effect depends on
Causes of Selection for Recombination S37
Figure 3: Typical dynamics of the Nee model. A, Linkage disequilibrium (LD; solid line) and epistasis (E; dashed line) of host population. B,
Immediate short-term effect (s0) and the delayed short-term effects s1 (dashed line) and s2 (dotted line). As an example of the immediate and delayed
short-term effects pointing in different directions, consider time step 9 (in A), where s0 ! 0 and s2 1 0 . The arrows point to the current and future
epistasis experienced as present by the host population at time step 9 (see “Results”). Parameters: sH p 0.2 ; sP p 1 ; rbaseline p 0.0025; Dr p 0.0005;
rmodifier p 0.5.
“future” epistasis, in the sense that the combination of
alleles created now will experience natural selection not
only in the current fitness landscape but also in the fitness
landscape present in n generations. Thus, starting from a
given value of LD at generation t, the immediate and delayed short-term effects can be qualitatively different only
if the sign of epistasis at generation t ⫹ n is different from
the sign of epistasis at generation t. As can be seen in
figure 3A, epistasis changes its sign only after a period
during which epistasis and LD are of the same sign (i.e.,
during which the immediate short-term effect selected
against recombination). Consequently, the two effects can
be qualitatively different only when the immediate shortterm effect is negative (selecting against recombination)
and the delayed short-term effect is positive.
We now turn our attention to the matching-allele
model. In contrast to the Nee model, the matching-allele
model exhibits directional selection and thus a potential
for a long-term effect. We find that the immediate shortterm effect (figs. 4C, 5C) is generally detrimental to re-
combination, in strong qualitative contrast to the observed
selection on the modifier (figs. 4A, 5A). It is, however,
quantitatively very weak and can thus be safely ignored.
These observations are independent of how strongly the
modifier is linked to the interaction loci. Interestingly, the
immediate short-term effect is appreciably strong only
when s P ≈ 1, which is consistent with previous findings
(Peters and Lively 2007).
In contrast, the delayed short-term effect (figs. 4F, 5F)
is generally in good agreement with selection on the modifier. Again, this observation is largely independent of how
strongly the modifier is linked to the interaction loci.
Finally, the long-term effect and the delayed short-term
effect are of about the same magnitude, but the direction
of the long-term effect depends strongly on the linkage
between the modifier and the interaction loci. When that
linkage is very strong, the long-term effect is detrimental
to recombination in areas of moderate selection and beneficial to recombination in areas of strong selection (fig.
4B). When the modifier is loosely linked, the opposite
S38
calculated by equation (10); E, sum of total short-term effect and long-term effect as calculated by equation (10); F, delayed short-term effect, given by D ⫺ C. Parameters: rbaseline p 0.0025;
Dr p 0.0005; rmodifier p rbaseline. Color coding as in figure 2.
Figure 4: Results from the matching-allele model. A, Change of modifier frequency; B, long-term effect; C, immediate short-term effect (measured as in fig. 2C); D, total short-term effect as
S39
Figure 5: Same as figure 4 but with a completely unlinked modifier locus; that is, rmodifier p 0.5. Color coding as in figure 2.
S40 The American Naturalist
pattern is observed: the long-term effect is beneficial to
recombination in areas of weak selection and detrimental
to recombination in areas of strong selection (fig. 5B).
Taken together, these observations suggest that, in general, the delayed short-term effect is the best qualitative
predictor for selection of recombination. The long-term
effect can support the delayed short-term effect, but even
if it causes selection against recombination, it is hardly
ever strong enough to override the effect of the delayed
short-term effect. This is particularly apparent in the case
of a completely unlinked modifier, where the long-term
effect and the delayed short-term effect result in almost
completely opposite selection pressures, with the longterm effect pointing in the wrong direction over almost
the entire parameter space.
Discussion
Overall, our simulations suggest that the real driving force
underlying the RQH is neither the immediate short-term
effect nor the long-term effect but, in fact, the delayed
short-term effect. As always, things turn out to be more
complicated when greater levels of detail are considered,
but in general, this statement holds, at least within the
realm of the simulations presented here.
In the Nee model, where a long-term effect can be excluded, the delayed short-term effect is a perfect qualitative
predictor for the evolution of recombination. In terms of
quantitative prediction, the contribution of the delayed
short-term effect is about an order of magnitude higher
than the immediate short-term effect. As we have argued
previously (Salathé et al. 2008b), equating the short-term
effect with the immediate short-term effect only is therefore problematic. As can be seen clearly in figure 2C, the
immediate short-term effect would predict selection
against higher levels of recombination in more than half
of the parameter space, while higher recombination is, in
fact, selected for in more than 90% of the parameter space.
Why is it that the immediate short-term effect is not a
good predictor for selection of recombination? It is worth
remembering that the short-term effect is the effect by
which recombination creates new combinations of alleles
that might be more or less fit than the combinations of
alleles in the previous generation. While this effect becomes manifest immediately in the next generation, it is
not confined to that generation only. A modifier associated
with new allelic combinations created by recombination
will also be subject to selection in future generations, that
is, with a delay. This delay is a fundamentally important
idiosyncrasy of the RQH: because of fluctuating epistasis,
the combinations of alleles that are unfit (or fit) in the
current generation might be fit (or unfit) in future generations. Figure 3B shows that the effect in future gen-
erations is, on average, more positive (dashed and dotted
lines) than the effect in the first generation of recombinant
offspring (solid line). Clearly, the delayed effects become
less important as the linkage between the modifier and the
interaction loci becomes weaker. We have shown previously, however, that even in the case of minimal linkage
(i.e., full recombination between the modifier and the interaction loci), the delayed effects outweigh the immediate
effect (see fig. Ib in box 2 of Salathé et al. 2008b).
The results of the matching-allele model show that the
discrepancy between the immediate and delayed shortterm effects is even more pronounced. Independent of the
linkage between modifier and interaction loci, the immediate short-term effect causes selection against recombination across almost the entire parameter space (figs.
4C, 5C), while the delayed short-term effect causes selection for recombination. However, because the immediate
short-term effect is one to two orders of magnitude weaker
than the delayed short-term effect, it is also largely irrelevant in determining the fate of a recombination modifier.
The delayed short-term effect appears to be a very good
qualitative proxy for selection on the modifier.
The long-term effect is of approximately the same magnitude as the delayed short-term effect. However, the pattern of the long-term effect depends strongly on the linkage between modifier and interaction loci. When this
linkage is strong (fig. 4), the long-term effect causes selection for recombination when at least one of the coevolving species experiences sufficient selection from their
interaction (with the exception of very weak selection on
both host and parasite). However, when linkage is loose
(fig. 5), the opposite pattern emerges: the long-term effect
then causes selection against recombination when at least
one of the coevolving species experiences sufficient selection from their interaction (with the notable exception of
very strong selection on both host and parasite). Thus, in
the case of loose linkage between modifier and interaction
loci, the long-term effect and the delayed short-term effect
exhibit roughly opposite patterns, but because the delayed
short-term effect is slightly stronger in this case, the qualitative outcome on selection on the modifier is predicted
better by the delayed short-term effect.
At first sight, it might seem that while the delayed shortterm effect is a better proxy than the long-term effect for
the evolution of recombination, its superiority is only marginal. However, this outcome strongly depends on which
area of the parameter space one is looking at. Little is
known about the effective fitness effects of parasites on
hosts and vice versa, but while there are some parasites
that have devastating effects on the fitness of their hosts
(e.g., castrating parasites), the majority of parasites probably exert much weaker selective pressure. If we zoom in
on the area where s H ≤ 0.2, a clearer picture emerges: the
Causes of Selection for Recombination S41
long-term effect and the delayed short-term effect are almost in perfect disagreement about the direction of selection they cause on the modifier, independent of the
linkage between modifier and interaction loci. The actual
direction of selection on the modifier, however, is always
in agreement with the delayed short-term effect: in no case
is the long-term effect strong enough to reverse the selection caused by the delayed short-term effect.
Our results are limited in three important ways. First,
the results obtained in this study are based on two specific
interaction models (the Nee model and the matching-allele
model). To what extent these results hold in other models
remains to be investigated. In the matching-allele model,
for example, the long-term effect is of only slightly weaker
magnitude than the delayed short-term effect. Thus, it
could very well be that a small departure from the strict
matching-allele model would affect the slightly skewed balance between the two effects and eventually reverse it.
Second, we have focused exclusively on the spread of a
recombination modifier. Recent work (Peters and Lively
2007) has shown that the short- and long-term effects
during the spread of a modifier can be very different from
the effects at equilibrium. Finally, our results are based on
relatively low recombination rates, an assumption that decreases the parameter regions where the conditions for
quasi-linkage equilibrium (QLE) are met (i.e., weak selection relative to recombination). For example, Otto and
Nuismer (2004) have shown that the long-term effect selects for recombination at QLE, indicating that the parameter region in figures 4B and 5B (where the long-term
effect selects for higher recombination near the origin)
would expand.
A continuing problem is that the terminology of shortand long-term effects is potentially confusing, particularly
in the context of the RQH. The short-term effect was
named to reflect the fact that it becomes manifest in the
generation immediately after a recombination event. But
since the short-term effect can also be felt many generations later (i.e., in the long term), it can easily be confused
with the long-term effect. At the same time, the shortterm effect can also be confused with an effect restricted
to the next generation only, which can also be misleading
because, as we have shown here, the effect in the generation
after a recombination event (the immediate short-term
effect) can select against recombination while the effect in
subsequent generations (the delayed short-term effect) can
select for recombination. In practice, this also means that
measuring the recombination load, given here by the immediate short-term effect, will not be sufficient to assess
the validity of the RQH. Because the delayed short-term
effect turns out to be crucial for the evolution of recombination in the RQH—at least in the interaction models
considered here—ignoring this effect can lead to the wrong
conclusions. Our results, therefore, highlight the importance of differentiating clearly between the immediate and
the delayed short-term effects when attempting to elucidate the mechanism underlying selection for recombination in the RQH.
Acknowledgments
We would like to thank S. Otto for the invitation to the
2008 Vice-Presidential Symposium of American Society of
Naturalists. Thanks also to C. Lively, D. Roze, and R. Salathé for helpful comments on the manuscript. This work
was supported by the Swiss National Science Foundation
and in part by National Institutes of Health grant
GM28016 to M. W. Feldman.
APPENDIX
The expressions for the short- and long-term effects (eq.
[10]) can be reformulated as a function of LD and allele
frequencies. First, a short calculation shows that
¯ p (w2 ⫺ w4 )(1 ⫺ p2 ) ⫹ (w1 ⫺ w3 )p2 p s 1,
⭸p 1w
¯ p (w3 ⫺ w4 )(1 ⫺ p1) ⫹ (w1 ⫺ w2 )p1 p s 2 ,
⭸p 2w
¯ p (w1 ⫹ w4 ) ⫺ (w2 ⫹ w3 ) p ␧add ,
⭸LDw
where the terms s1 and s2 correspond to the selection on
the interaction loci 1 and 2, respectively. Furthermore, the
conditional allele frequencies (pim, piM) and LDs (LDM,
LDm) can be expressed in terms of allele frequencies (p1,
p2, and pM), two-way LD (D12, D1M, D2M), and three-way
LD (D12M; Barton 1995; Shpak and Gavrilets 2006). With
these transformations, the long- and short-term effects are
s long p
D1M s 1 ⫹ D2M s 2
¯
p M(1 ⫺ p M)w
(A1)
and
sshort p
D12M p M(1 ⫺ p M) ⫺ D1MD2M(1 ⫺ 2p M)
␧add ,
¯
[p M(1 ⫺ p M)]2w
(A2)
respectively. Following Barton (1995), an exact expression
for the long- and short-term effects can be derived as
follows. The total selection acting on the modifier, sM (see
eq. [8]) can be expressed in terms of allele frequencies and
LDs:
sM p
D1M s 1 ⫹ D2M s 2
D12M ␧add
⫹
.
¯
¯
p M(1 ⫺ p M)w
pm(1 ⫺ pm)w
(A3)
Thus, sM is the sum of the long-term effect, which results
S42 The American Naturalist
from the association of the modifier with individual alleles
(D1M and D2M), and the short-term effect, which results
from the association of the modifier with allele combinations (D12M). Comparing this exact derivation of the
long- and short-term effects with the corresponding terms
of the linear approximation (eqq. [A1], [A2]), we see that
the exact calculation and the linear approximation of the
long-term effect are identical, while the exact calculation
and the linear approximation of the short-term effect differ
by a term proportional to D1MD2M. Given the properties
of the linear approximation, this term is negligible for weak
modifiers (a result we confirmed by numerical simulation).
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